\chapter{Summary of MoM terms and formulations}
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\par
\renewcommand{\arraystretch}{2}
\begin{table}[h!]
\centering
\begin{tabular}{|c||c|c|}
  \hline
  \multicolumn{3}{|c|}{Integral equations for a PEC body} \\
  \hline
  \hline
   & EFIE & MFIE \\
  \hline
  outside & $\field{E}^\text{inc} =  - \frac{1}{j \omega \varepsilon_1} \operator{D}_1 \left( \current{J}_{S}\right)$ & $\field{H}^\text{inc} = - \frac{1}{2} \vect{\hat{n}} \times \current{J}_{S} - \operator{K}_1 \left(\current{J}_{S}\right)$ \\
  \hline
  \hline
  \multicolumn{3}{|c|}{Integral equations for a dielectric body} \\
  \hline
  \hline
   & EFIE & MFIE \\
  \hline
  outside & $\field{E}^\text{inc} =  - \frac{1}{j \omega \varepsilon_1} \operator{D}_1 \left( \current{J}_{S}\right) + \frac{1}{2} \vect{\hat{n}} \times \current{M}_{S} + \operator{K}_1 \left(\current{M}_{S}\right)$ & $\field{H}^\text{inc} = - \frac{1}{2} \vect{\hat{n}} \times \current{J}_{S} - \operator{K}_1 \left(\current{J}_{S}\right) - \frac{1}{j \omega \mu_1} \operator{D}_1 \left( \current{M}_{S}\right)$ \\
  \hline
  inside  & $0 = \frac{1}{j \omega \varepsilon_2} \operator{D}_2\left(\current{J}_{S}\right) + \frac{1}{2} \vect{\hat{n}} \times \current{M}_{S} -  \operator{K}_2\left( \current{M}_{S}\right)$ & $0 = - \frac{1}{2} \vect{\hat{n}} \times \current{J}_{S} +  \operator{K}_2\left( \current{J}_{S}\right) + \frac{1}{j \omega \mu_2} \operator{D}_2\left(\current{M}_{S}\right) $ \\
  \hline
  \hline
 & $\operator{D}_i\left(\current{X}\right) = \left(\nabla \nabla \cdot + k_i^2\right) \int_{S} G_i\left(\vect{r}, \vect{r}'\right) \current{X}\left(\vect{r}'\right) d\vect{r}'$  & $\operator{K}_i\left(\current{X}\right) = \int_S \nabla G_i\left(\vect{r}, \vect{r}'\right) \times \current{X}\left(\vect{r}'\right) d\vect{r}'$ \\
  \hline
\end{tabular}
\end{table}

\begin{table}[h!]
\centering
\begin{tabular}{|c||c|c|}
  \hline
  \multicolumn{3}{|c|}{Method of moments discretization for PEC} \\
  \hline
  \hline
   & MoM EFIE & MoM MFIE \\
  \hline
  outside (sign changed) & $\underline{V}^E =  \frac{1}{j \omega \varepsilon_1}  \underline{\underline{D}}^{(1)} \underline{I}$ & $\underline{V}^H =  \left[\underline{\underline{J}} + \underline{\underline{K}}^{(1)} \right] \underline{I}$ \\
  \hline
  \hline
  \multicolumn{3}{|c|}{Method of moments discretization for dielectric} \\
  \hline
  \hline
   & MoM EFIE & MoM MFIE \\
  \hline
  outside (sign changed) & $\underline{V}^E =  \frac{1}{j \omega \varepsilon_1}  \underline{\underline{D}}^{(1)} \underline{I} -\left[\underline{\underline{J}} + \underline{\underline{K}}^{(1)} \right] \underline{M} $ & $\underline{V}^H =  \left[\underline{\underline{J}} + \underline{\underline{K}}^{(1)} \right] \underline{I} + \frac{1}{j \omega \mu_1}  \underline{\underline{D}}^{(1)} \underline{M}$ \\
  \hline
  inside  & $\underline{0} =  \frac{1}{j \omega \varepsilon_2}  \underline{\underline{D}}^{(2)} \underline{I} + \left[\underline{\underline{J}} - \underline{\underline{K}}^{(2)} \right] \underline{M} $ & $\underline{0} =  \left[-\underline{\underline{J}} + \underline{\underline{K}}^{(2)} \right] \underline{I} + \frac{1}{j \omega \mu_2}  \underline{\underline{D}}^{(2)} \underline{M}$ \\
  \hline
  \hline
$D_{mn}^{(i)}$ & \multicolumn{2}{|c|}{$\int_{D_m}\vect{g}_m \arg{\vect{r}} \cdot \left( \left(\nabla \nabla \cdot + k_i^2\right) \int_{D_n} G_i\left(\vect{r}, \vect{r}'\right) \current{f}_n\left(\vect{r}'\right) d\vect{r}' \right) d\vect{r}$} \\
\hline
 $K_{mn}^{(i)}$ & \multicolumn{2}{|c|}{$\int_{D_m} \vect{g}_m \arg{\vect{r}} \cdot \left(\int_{D_n} \nabla G_i\left(\vect{r}, \vect{r}'\right) \times \current{f}_n\left(\vect{r}'\right) d\vect{r}'\right) d\vect{r}$} \\
  \hline
$J_{mn}$ & \multicolumn{2}{|c|}{$\int_{D_m} \frac{1}{2} \vect{g}_m \arg{\vect{r}} \cdot \vect{\hat{n}} \times \current{f}_{n}\arg{\vect{r}}d\vect{r}$} \\
  \hline
$V_m^{E,H}$ & \multicolumn{2}{|c|}{$-\int_{D_m}\vect{g}_m \arg{\vect{r}} \cdot \left(\field{E}^\text{inc},\field{H}^\text{inc}\right) d\vect{r}$} \\
\hline
\end{tabular}
\end{table}


